Question

Total blood volume (in ml) per body weight (in kg) is important in medical research. For healthy adults, the red blood cell volume mean is about μ = 28 ml/kg.† Red blood cell volume that is too low or too high can indicate a medical problem. Suppose that Roger has had seven blood tests, and the red blood cell volumes were as follows.

34

26

42

35

28

37

31

The sample mean is x ≈ 33.3 ml/kg. Let x be a random variable that represents Roger's red blood cell volume. Assume that x has a normal distribution and σ = 4.75. Do the data indicate that Roger's red blood cell volume is different (either way) from μ = 28 ml/kg? Use a 0.01 level of significance.

(a) What is the level of significance?


State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H₀: μ = 28 ml/kg; H₁: μ < 28 ml/kg; left-tailedH₀: μ ≠ 28 ml/kg; H₁: μ = 28 ml/kg; two-tailed    H₀: μ = 28 ml/kg; H₁: μ > 28 ml/kg; right-tailedH₀: μ = 28 ml/kg; H₁: μ ≠ 28 ml/kg; two-tailed

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since we assume that x has a normal distribution with known σ.The Student's t, since n is large with unknown σ.    The standard normal, since we assume that x has a normal distribution with known σ.The standard normal, since we assume that x has a normal distribution with unknown σ.

Compute the z value of the sample test statistic. (Round your answer to two decimal places.)


(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

Answer 1

The statistic software output for this problem is:

One sample Z summary hypothesis test: : Mean of population Ho : u = 28 HA: 28 Standard deviation = 4.75 Hypothesis test results: Mean n Sample Mean Std. Err. Z-Stat 33.3 1.7953312 2.9521015 P-value 0.0032

(a)

the level of significance

two-tailed test

H₀ : μ = 28
H_A : μ ≠ 28

(b)

The standard normal, since we assume that x has a normal distribution with known σ

z = 2.95

(c)

P-value = 0.0032

Graph III

(d)

At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant